# Isomorphic and isometric structure of the optimal domains for Hardy-type operators

## TL;DR

This paper explores the structure of optimal domains for Hardy-type operators, revealing their close relation to classical function spaces like L^1 and L^, and investigates their properties and applications in fixed point theory.

## Contribution

It characterizes the isomorphic and isometric structures of optimal Hardy-type operator domains and examines their properties, including non-reflexivity and non-isomorphism to dual spaces.

## Key findings

- Cesro and Copson spaces contain complemented copies of L^1
- Generalized Tandori spaces resemble L^ and contain isometric copies of 
- Cesro construction does not commute with measure support truncation

## Abstract

We investigate structure of the optimal domains for the Hardy-type operators including, for example, the classical Ces\`aro, Copson and Volterra operators as well as for some of their generalizations. We prove that, in some sense, the abstract Ces\`aro and Copson function spaces are closely related to the space $L^1$, namely, they contain "in the middle" a complemented copy of $L^1[0,1]$, asymptotically isometric copy of $\ell^1$ and also can be renormed to contain an isometric copy of $L^1[0,1]$. Moreover, the generalized Tandori function spaces are quite similar to $L^\infty$ because they contain an isometric copy of $\ell^\infty$ and can be renormed to contain an isometric copy of $L^\infty[0,1]$. Several applications to the metric fixed point theory will be given. Next, we prove that the Ces\`aro construction $X \mapsto CX$ does not commutate with the truncation operation of the measure space support. We also study whether a given property transfers between a Banach function space $X$ and the space $TX$, where $T$ is the Ces\`aro or the Copson operator. In particular, we find a large class of properties which do not lift from $TX$ into $X$ and prove that the abstract Ces\`aro and Copson function spaces are never reflexive, are not isomorphic to a dual space and do not have the Radon--Nikodym property in general.